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G = (C2×C10)⋊D8order 320 = 26·5

3rd semidirect product of C2×C10 and D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C10)⋊3D8, C4⋊D43D5, C52C822D4, C53(C87D4), C4⋊C4.58D10, C10.55(C2×D8), C4.170(D4×D5), C207D423C2, C221(D4⋊D5), (C2×D4).38D10, (C2×C20).263D4, C20.147(C2×D4), D206C435C2, C10.97(C4○D8), D4⋊Dic515C2, C10.D836C2, (C22×C10).84D4, C20.183(C4○D4), C4.59(D42D5), C10.93(C4⋊D4), (C2×C20).357C23, (D4×C10).54C22, (C22×C4).340D10, C23.39(C5⋊D4), (C2×D20).101C22, C4⋊Dic5.142C22, C2.14(Dic5⋊D4), C2.16(D4.8D10), (C22×C20).161C22, (C2×D4⋊D5)⋊10C2, (C5×C4⋊D4)⋊3C2, C2.10(C2×D4⋊D5), (C22×C52C8)⋊3C2, (C2×C10).488(C2×D4), (C2×C4).105(C5⋊D4), (C5×C4⋊C4).105C22, (C2×C4).457(C22×D5), C22.163(C2×C5⋊D4), (C2×C52C8).256C22, SmallGroup(320,665)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×C10)⋊D8
C1C5C10C20C2×C20C2×D20C207D4 — (C2×C10)⋊D8
C5C10C2×C20 — (C2×C10)⋊D8
C1C22C22×C4C4⋊D4

Generators and relations for (C2×C10)⋊D8
 G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, ac=ca, dad=ab5, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 550 in 134 conjugacy classes, 45 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C5, C8 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], D5, C10 [×3], C10 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×4], D8 [×2], C22×C4, C2×D4, C2×D4 [×3], Dic5, C20 [×2], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], D4⋊C4 [×2], C2.D8, C4⋊D4, C4⋊D4, C22×C8, C2×D8, C52C8 [×2], C52C8, D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×3], C5×D4 [×4], C22×D5, C22×C10, C22×C10, C87D4, C2×C52C8 [×2], C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, D4⋊D5 [×2], C5×C22⋊C4, C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C10.D8, D206C4, D4⋊Dic5, C22×C52C8, C207D4, C2×D4⋊D5, C5×C4⋊D4, (C2×C10)⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C4○D8, C5⋊D4 [×2], C22×D5, C87D4, D4⋊D5 [×2], D4×D5, D42D5, C2×C5⋊D4, C2×D4⋊D5, Dic5⋊D4, D4.8D10, (C2×C10)⋊D8

Smallest permutation representation of (C2×C10)⋊D8
On 160 points
Generators in S160
(1 98)(2 99)(3 100)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 58 38 90 30 70 50 78)(2 57 39 89 21 69 41 77)(3 56 40 88 22 68 42 76)(4 55 31 87 23 67 43 75)(5 54 32 86 24 66 44 74)(6 53 33 85 25 65 45 73)(7 52 34 84 26 64 46 72)(8 51 35 83 27 63 47 71)(9 60 36 82 28 62 48 80)(10 59 37 81 29 61 49 79)(11 105 145 125 153 93 133 113)(12 104 146 124 154 92 134 112)(13 103 147 123 155 91 135 111)(14 102 148 122 156 100 136 120)(15 101 149 121 157 99 137 119)(16 110 150 130 158 98 138 118)(17 109 141 129 159 97 139 117)(18 108 142 128 160 96 140 116)(19 107 143 127 151 95 131 115)(20 106 144 126 152 94 132 114)
(2 10)(3 9)(4 8)(5 7)(11 150)(12 149)(13 148)(14 147)(15 146)(16 145)(17 144)(18 143)(19 142)(20 141)(21 29)(22 28)(23 27)(24 26)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 50)(39 49)(40 48)(51 75)(52 74)(53 73)(54 72)(55 71)(56 80)(57 79)(58 78)(59 77)(60 76)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 90)(91 100)(92 99)(93 98)(94 97)(95 96)(101 104)(102 103)(105 110)(106 109)(107 108)(111 122)(112 121)(113 130)(114 129)(115 128)(116 127)(117 126)(118 125)(119 124)(120 123)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)

G:=sub<Sym(160)| (1,98)(2,99)(3,100)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,58,38,90,30,70,50,78)(2,57,39,89,21,69,41,77)(3,56,40,88,22,68,42,76)(4,55,31,87,23,67,43,75)(5,54,32,86,24,66,44,74)(6,53,33,85,25,65,45,73)(7,52,34,84,26,64,46,72)(8,51,35,83,27,63,47,71)(9,60,36,82,28,62,48,80)(10,59,37,81,29,61,49,79)(11,105,145,125,153,93,133,113)(12,104,146,124,154,92,134,112)(13,103,147,123,155,91,135,111)(14,102,148,122,156,100,136,120)(15,101,149,121,157,99,137,119)(16,110,150,130,158,98,138,118)(17,109,141,129,159,97,139,117)(18,108,142,128,160,96,140,116)(19,107,143,127,151,95,131,115)(20,106,144,126,152,94,132,114), (2,10)(3,9)(4,8)(5,7)(11,150)(12,149)(13,148)(14,147)(15,146)(16,145)(17,144)(18,143)(19,142)(20,141)(21,29)(22,28)(23,27)(24,26)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,75)(52,74)(53,73)(54,72)(55,71)(56,80)(57,79)(58,78)(59,77)(60,76)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,90)(91,100)(92,99)(93,98)(94,97)(95,96)(101,104)(102,103)(105,110)(106,109)(107,108)(111,122)(112,121)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151)>;

G:=Group( (1,98)(2,99)(3,100)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,58,38,90,30,70,50,78)(2,57,39,89,21,69,41,77)(3,56,40,88,22,68,42,76)(4,55,31,87,23,67,43,75)(5,54,32,86,24,66,44,74)(6,53,33,85,25,65,45,73)(7,52,34,84,26,64,46,72)(8,51,35,83,27,63,47,71)(9,60,36,82,28,62,48,80)(10,59,37,81,29,61,49,79)(11,105,145,125,153,93,133,113)(12,104,146,124,154,92,134,112)(13,103,147,123,155,91,135,111)(14,102,148,122,156,100,136,120)(15,101,149,121,157,99,137,119)(16,110,150,130,158,98,138,118)(17,109,141,129,159,97,139,117)(18,108,142,128,160,96,140,116)(19,107,143,127,151,95,131,115)(20,106,144,126,152,94,132,114), (2,10)(3,9)(4,8)(5,7)(11,150)(12,149)(13,148)(14,147)(15,146)(16,145)(17,144)(18,143)(19,142)(20,141)(21,29)(22,28)(23,27)(24,26)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,75)(52,74)(53,73)(54,72)(55,71)(56,80)(57,79)(58,78)(59,77)(60,76)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,90)(91,100)(92,99)(93,98)(94,97)(95,96)(101,104)(102,103)(105,110)(106,109)(107,108)(111,122)(112,121)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151) );

G=PermutationGroup([(1,98),(2,99),(3,100),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,81),(18,82),(19,83),(20,84),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,58,38,90,30,70,50,78),(2,57,39,89,21,69,41,77),(3,56,40,88,22,68,42,76),(4,55,31,87,23,67,43,75),(5,54,32,86,24,66,44,74),(6,53,33,85,25,65,45,73),(7,52,34,84,26,64,46,72),(8,51,35,83,27,63,47,71),(9,60,36,82,28,62,48,80),(10,59,37,81,29,61,49,79),(11,105,145,125,153,93,133,113),(12,104,146,124,154,92,134,112),(13,103,147,123,155,91,135,111),(14,102,148,122,156,100,136,120),(15,101,149,121,157,99,137,119),(16,110,150,130,158,98,138,118),(17,109,141,129,159,97,139,117),(18,108,142,128,160,96,140,116),(19,107,143,127,151,95,131,115),(20,106,144,126,152,94,132,114)], [(2,10),(3,9),(4,8),(5,7),(11,150),(12,149),(13,148),(14,147),(15,146),(16,145),(17,144),(18,143),(19,142),(20,141),(21,29),(22,28),(23,27),(24,26),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,50),(39,49),(40,48),(51,75),(52,74),(53,73),(54,72),(55,71),(56,80),(57,79),(58,78),(59,77),(60,76),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,90),(91,100),(92,99),(93,98),(94,97),(95,96),(101,104),(102,103),(105,110),(106,109),(107,108),(111,122),(112,121),(113,130),(114,129),(115,128),(116,127),(117,126),(118,125),(119,124),(120,123),(131,160),(132,159),(133,158),(134,157),(135,156),(136,155),(137,154),(138,153),(139,152),(140,151)])

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A···8H10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222222444444558···810···10101010101010101020···2020202020
size11112284022228402210···102···2444488884···48888

50 irreducible representations

dim111111112222222222224444
type+++++++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D8D10D10D10C4○D8C5⋊D4C5⋊D4D4×D5D42D5D4⋊D5D4.8D10
kernel(C2×C10)⋊D8C10.D8D206C4D4⋊Dic5C22×C52C8C207D4C2×D4⋊D5C5×C4⋊D4C52C8C2×C20C22×C10C4⋊D4C20C2×C10C4⋊C4C22×C4C2×D4C10C2×C4C23C4C4C22C2
# reps111111112112242224442244

Matrix representation of (C2×C10)⋊D8 in GL6(𝔽41)

4000000
0400000
001000
000100
000009
0000320
,
34350000
700000
001000
000100
0000400
0000040
,
34350000
870000
000700
00352400
00002912
00002929
,
34350000
870000
001000
00214000
000010
0000040

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,9,0],[34,7,0,0,0,0,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,0,35,0,0,0,0,7,24,0,0,0,0,0,0,29,29,0,0,0,0,12,29],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,21,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

(C2×C10)⋊D8 in GAP, Magma, Sage, TeX

(C_2\times C_{10})\rtimes D_8
% in TeX

G:=Group("(C2xC10):D8");
// GroupNames label

G:=SmallGroup(320,665);
// by ID

G=gap.SmallGroup(320,665);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^8=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^5,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽